Notes

In classical physics quantities like energy were always assumed to correspond to continuous variables. But in 1900 Max Planck noticed that fits to the measured spectrum of electromagnetic radiation produced by hot objects could be explained if there were discrete quanta of electromagnetic energy. And by 1910 work by Albert Einstein, notably on the photoelectric effect and on heat capacities of solids, had given evidence for discrete quanta of energy in both light and matter. In 1913 Niels Bohr then made the suggestion that the discrete spectrum of light emitted by hydrogen atoms could be explained as being produced by electrons making transitions between orbits with discrete quantized angular momenta. By 1920 ideas from celestial mechanics had been used to develop a formalism for quantized orbits which successfully explained various features of atoms and chemical elements. But it was not clear how to extend the formalism say to a problem like propagation of light through a crystal. In 1925, however, Werner Heisenberg suggested a new and more general formalism that became known as matrix mechanics. The original idea was to imagine describing the state of an atom in terms of an array of amplitudes for virtual oscillators with each possible frequency. Particular conditions amounting to quantization were then imposed on matrices of transitions between these, and the idea was introduced that only certain kinds of amplitude combinations could ever be observed. In 1923 Louis de Broglie had suggested that just as light—which in optics was traditionally described in terms of waves—seemed in some respects to act like discrete particles, so conversely particles like electrons might in some respects act like waves. In 1926 Erwin Schrödinger then suggested a partial differential equation for the wave functions of particles like electrons. And when effectively restricted to a finite region, this equation allowed only certain modes, corresponding to discrete quantum states—whose properties turned out to be exactly the same as implied by matrix mechanics. In the late 1920s Paul Dirac developed a more abstract operator-based formalism. And by the end of the 1920s basic practical quantum mechanics was established in more or less the form it appears in textbooks today. In the period since, increasing computational capabilities have allowed coupled Schrödinger equations for progressively more particles to be solved (reasonably accurate solutions for hundreds of particles can now be found), allowing ever larger studies in atomic, molecular, nuclear and solid-state physics. A notable theoretical interest starting in the 1980s was so-called quantum chaos, in which it was found that modes (wave functions) in regions like stadiums that did not yield simple analytical solutions tended to show complicated and seemingly random forms.

Basic quantum mechanics is set up to describe how fixed numbers of particles behave—say in externally applied electromagnetic or other fields. But to describe things like fields one must allow particles to be created and destroyed. In the mid-1920s there was already discussion of how to set up a formalism for this, with an underlying idea again being to think in terms of virtual oscillators—but now one for each possible state of each possible one of any number of particles. At first this was just applied to a pure electromagnetic field of non-interacting photons, but by the end of the 1920s there was a version of quantum electrodynamics (QED) for interacting photons and electrons that is essentially the same as today. To find predictions from this theory a so-called perturbation expansion was made, with successive terms representing progressively more interactions, and each having a higher power of the so-called coupling constant α≃1/137. It was immediately noticed, however, that self-interactions of particles would give rise to infinities, much as in classical electromagnetism. At first attempts were made to avoid this by modifying the basic theory (see page 1044). But by the mid-1940s detailed calculations were being done in which infinite parts were just being dropped—and the results were being found to agree rather precisely with experiments. In the late 1940s this procedure was then essentially justified by the idea of renormalization: that since in all possible QED processes only three different infinities can ever appear, these can in effect systematically be factored out from all predictions of the theory. Then in 1949 Feynman diagrams were introduced (see note below) to represent terms in the QED perturbation expansion—and the rules for these rapidly became what defined QED in essentially all practical applications. Evaluating Feynman diagrams involved extensive algebra, and indeed stimulated the development of computer algebra (including my own interest in the field). But by the 1970s the dozen or so standard processes discussed in QED had been calculated to order α2—and by the mid-1980s the anomalous magnetic moment of the electron had been calculated to order α4, and nearly one part in a trillion (see note below).

But despite the success of perturbation theory in QED it did not at first seem applicable to other issues in particle physics. The weak interactions involved in radioactive beta decay seemed too weak for anything beyond lowest order to be relevant—and in any case not renormalizable. And the strong interactions responsible for holding nuclei together (and associated for example with exchange of pions and other mesons) seemed too strong for it to make sense to do an expansion with larger numbers of individual interactions treated as less important. So this led in the 1960s to attempts to base theories just on setting up simple mathematical constraints on the overall so-called S matrix defining the mapping from incoming to outgoing quantum states. But by the end of the 1960s theoretical progress seemed blocked by basic questions about functions of several complex variables, and predictions that were made did not seem to work well.

By the early 1970s, however, there was increasing interest in so-called gauge or Yang–Mills theories formed in essence by generalizing QED to operate not just with a scalar charge, but with charges viewed as elements of non-Abelian groups. In 1972 it was shown that spontaneously broken gauge theories of the kind needed to describe weak interactions were renormalizable—allowing meaningful use of perturbation theory and Feynman diagrams. And then in 1973 it was discovered that QCD—the gauge theory for quarks and gluons with SU(3) color charges—was asymptotically free (it was known to be renormalizable), so that for processes probing sufficiently small distances, its effective coupling was small enough for perturbation theory. By the early 1980s first-order calculations of most basic QCD processes had been done—and by the 1990s second-order corrections were also known. Schemes for adding up all Feynman diagrams with certain very simple repetitive or other structures were developed. But despite a few results about large-distance analogs of renormalizability, the question of what QCD might imply for processes at larger distances could not really be addressed by such methods.

In 1941 Richard Feynman pointed out that amplitudes in quantum theory could be worked out by using path integrals that sum with appropriate weights contributions from all possible histories of a system. (The Schrödinger equation is like a diffusion equation in imaginary time, so the path integral for it can be thought of as like an enumeration of random walks. The idea of describing random walks with path integrals was discussed from the early 1900s.) At first the path integral was viewed mostly as a curiosity, but by the late 1970s it was emerging as the standard way to define a quantum field theory. Attempts were made to see if the path integral for QCD (and later for quantum gravity) could be approximated with a few exact solutions (such as instantons) to classical field equations. By the early 1980s there was then extensive work on lattice gauge theories in which the path integral (in Euclidean space) was approximated by randomly sampling discretized field configurations. But—I suspect for reasons that I discuss in the note below—such methods were never extremely successful. And the result is that beyond perturbation theory there is still no real example of a definitive success from standard relativistic quantum field theory. (In addition, even efforts in the context of so-called axiomatic field theory to set up mathematically rigorous formulations have run into many difficulties—with the only examples satisfying all proposed axioms typically in the end being field theories without any real interactions. In condensed matter physics there are nevertheless cases like the Kondo model where exact solutions have been found, and where the effective energy function for electrons happens to be roughly the same as in a relativistic theory.)

As mentioned on page 1044, ordinary quantum field theory in effect deals only with point particles. And indeed a recurring issue in it has been difficulty with constraints and redundant degrees of freedom—such as those associated with extended objects. (A typical goal is to find variables in which one can carry out what is known as canonical quantization: essentially applying the same straightforward transformation of equations that happens to work in ordinary elementary quantum mechanics.) One feature of string theory and its generalizations is that they define presumably consistent quantum field theories for excitations of extended objects—though an analog of quantum field theory in which whole strings can be created and destroyed has not yet been developed.

When the formalism of quantum mechanics was developed in the mid-1920s there were immediately questions about its interpretation. But it was quickly suggested that given a wave function ψ from the Schrödinger equation Abs[ψ]2 should represent probability—and essentially all practical applications have been based on this ever since. From a conceptual point of view it has however often seemed peculiar that a supposedly fundamental theory should talk only about probabilities. Following the introduction of the uncertainty principle and related formalism in the 1920s one idea that arose was that—in rough analogy to relativity theory—it might just be that there are only certain quantities that are observable in definite ways. But this was not enough, and by the 1930s it was being suggested that the validity of quantum mechanics might be a sign that whole new general frameworks for philosophy or logic were needed—a notion supported by the apparent need to bring consciousness into discussions about measurement in quantum mechanics (see page 1063). The peculiar character of quantum mechanics was again emphasized by the idealized experiment of Albert Einstein, Boris Podolsky and Nathan Rosen in 1935. But among most physicists the apparent lack of an ordinary mechanistic way to think about quantum mechanics ended up just being seen as another piece of evidence for the fundamental role of mathematical formalism in physics.

One way for probabilities to appear even in deterministic systems is for there to be hidden variables whose values are unknown. But following mathematical work in the early 1930s it was usually assumed that this could not be what was going on in quantum mechanics. In 1952 David Bohm did however manage to construct a somewhat elaborate model based on hidden variables that gave the same results as ordinary quantum mechanics—though involved infinitely fast propagation of information. In the early 1960s John Bell then showed that in any hidden variables theory of a certain general type there are specific inequalities that combinations of probabilities must satisfy (see page 1064). And by the early 1980s experiments had shown that such inequalities were indeed violated in practice—so that there were in fact correlations of the kind suggested by quantum mechanics. At first these just seemed like isolated esoteric effects, but by the mid-1990s they were being codified in the field of quantum information theory, and led to constructions with names like quantum cryptography and quantum teleportation.

Particularly when viewed in terms of path integrals the standard formalism of quantum theory tends to suggest that quantum systems somehow do more computation in their evolution than classical ones. And after occasional discussion as early as the 1950s, this led by the late 1980s to extensive investigation of systems that could be viewed as quantum analogs of idealized computers. In the mid-1990s efficient procedures for integer factoring and a few other problems were suggested for such systems, and by the late 1990s small experiments on these were beginning to be done in various types of physical systems. But it is becoming increasingly unclear just how the idealizations in the underlying model really work, and to what extent quantum mechanics is actually in the end even required—as opposed, say, just to classical wave phenomena. (See page 1147.)

Partly as a result of discussions about measurement there began to be questions in the 1980s about whether ordinary quantum mechanics can describe systems containing very large numbers of particles. Experiments in the 1980s and 1990s on such phenomena as macroscopic superposition and Bose–Einstein condensation nevertheless showed that standard quantum effects still occur with trillions of atoms. But inevitably the kinds of general phenomena that I discuss in this book will also occur—leading to all sorts of behavior that at least cannot readily be foreseen just from the basic rules of quantum mechanics.